Multi-functional reconfigurable intelligence surface integrating signal reflection, refraction and amplification and energy harvesting and application thereof

Abstract

A multi-functional reconfigurable intelligence surface (MF-RIS) integrating signal reflection, refraction and amplification and energy harvesting and an application thereof are provided. The MF-RIS can support wireless signal reflection, refraction and amplification and energy harvesting on one surface, to amplify, reflect, or refract a signal through harvested energy, and further enhance effective coverage of wireless signals. When a signal model of the MF-RIS constructed in the present disclosure is applied to a multi-user wireless network, a non-convex optimization problem of jointly designing operation modes and parameters that include BS transmit beamforming, and different components and a deployment position of the MF-RIS is constructed with an objective of maximizing a sum rate (SR) of a plurality of users in an MF-RIS-assisted non-orthogonal multiple access network. Then, an iterative optimization algorithm is designed to effectively solve the non-convex optimization problem, to maximize the SR of the plurality of users.

Claims

1. A multi-functional reconfigurable intelligence surface (MF-RIS) integrating signal reflection, refraction and amplification and energy harvesting, having two working modes, namely an energy harvesting mode and a signal relay mode, wherein in the signal relay mode, an incident signal is reflected and refracted through surface equivalent electrical impedance and magnetoimpedance elements, the incident signal is divided into a first part and a second part by controlling electric current and magnetic current through a microcontroller unit (MCU) chip, the first part is reflected to reflection half-space and the second part is refracted to refraction half-space, and a reflected signal and a refracted signal are amplified through an amplifier circuit; in the energy harvesting mode, radio frequency (RF) energy is obtained from the incident signal and converted into direct current (DC) power through an impedance matcher, an RF-DC conversion circuit and a capacitor, and an energy management module controls energy to be stored in an energy storage apparatus or supplied for operation of a phase shifter and the amplifier circuit; and a circuit connection is adjusted such that each element is flexibly switched between the energy harvesting mode and the signal relay mode.

2. An application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting according to claim 1 in a multi-user wireless network, comprising the following steps: S1: designing operation modes and parameters, comprising base station (BS) transmit beamforming, and different components and a deployment position of the MF-RIS, and constructing a mixed integer non-linear programming non-convex optimization problem and constraints with an objective of maximizing an achievable sum rate (SR) of all users; S2: decomposing the non-convex problem constructed in S1 into three subproblems: a BS transmit beamforming optimization problem, an MF-RIS coefficient design problem, and an MF-RIS deployment optimization problem; and S3: performing alternating optimization (AO) on the subproblems obtained through decomposition in S2 to ensure that each sub-algorithm converges to a local optimum, which comprises: for the BS transmit beamforming optimization problem, introducing auxiliary variables and solving the BS transmit beamforming optimization problem through a sequential rank-one constraint relaxation (SROCR) method; for the MF-RIS coefficient design problem, introducing an auxiliary variable, replacing a non-convex objective function with a convex upper bound (CUB) of the non-convex objective function, processing an equality constraint through a penalty function method, and designing a coefficient of the MF-RIS; and for the MF-RIS deployment optimization problem, designing the position of the MF-RIS through a local area optimization method, and processing a non-convex term through successive convex approximation (SCA) to transform the MF-RIS deployment optimization problem into a solvable convex problem.

3. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 2, wherein the optimization problem and the constraints in S1 are as follows: $\underset{f_k,{}_k,w}{\max }{.Math.}_k{.Math.}_{j{J}_k}{R}_{j.fwdarw.j}^k$ $s.t.$ ${.Math.}_k{.Math.f_k.Math.}^2{P}_{BS}^{m\mathrm{ax}},$ ${}_k{R}_{\mathrm{MF}},$ $kK,$ $R_{j.fwdarw.j}^k{R}_{\mathrm{kj}}^{min},$ $kK,$ $j{J}_k,$ $wP=\left\{{\left[x,y,z\right]}^T|x_{min}x{x}_{\mathrm{ma}x},y_{min}y{y}_{\mathrm{ma}x},z_{min}z{z}_{\mathrm{ma}x}\right\},$ $\frac{{.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_k.Math.}^2}{{.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_{\stackrel{}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2}\frac{{.Math.{\stackrel{}{h}}_{kl}{f}_k.Math.}^2}{{.Math.{\stackrel{}{h}}_{kl}{f}_{\stackrel{}{k}}.Math.}^2+{.Math.g_{kl}^H{}_k{n}_s.Math.}^2+{}_u^2},$ $kK,$ $j{J}_k,$ $l{L}_k,$ $R_{l.fwdarw.j}^k{R}_{j.fwdarw.j}^k,$ $kK,$ $j{J}_k,$ $l{L}_k$ $2\left(P_b+P_{DC}\right){.Math.}_m{}_m+\left(M-{.Math.}_m{}_m\right)P_C+P_O{.Math.}_m{P}_m^A,$ wherein a quantity of elements of the MF-RIS is M, a set of the elements of the MF-RIS is indexed as M={1, . . . , M}, and m is an m.sup.th element in the set of the elements; a quantity of antennas is J, a space set is K={r,t}, a user set is J={1, 2, . . . , J}, j is a j.sup.th element in the user set, J.sub.k={1, 2, . . . , J.sub.k} represents a user set in space K, J.sub.rJ.sub.t=J, and if k=t, k=r; if k=r, k=t, and a user j in the space K is indexed as U.sub.kj; h.sub.kj=h.sub.kj.sup.H+g.sub.kj.sup.H.sub.kH represents a combined channel vector from a BS to the user U.sub.kj, .sub.k is the coefficient of the MF-RIS, and R.sub.MF is a feasible coefficient set of the MF-RIS; R.sub.j.fwdarw.j.sup.k is an achievable rate of an expected signal of the user U.sub.kj, R.sub.kj.sup.min represents a minimum quality of service requirement of the user U.sub.kj, f.sub.k is a transmit beamforming vector of the space k, P.sub.BS.sup.max represents maximum transmit power of the BS, P represents a predefined deployment area of the MF-RIS, considering a three-dimensional (3D) Cartesian coordinate system, positions of the BS, the MF-RIS, and the user U.sub.kj are respectively w.sub.b=[x.sub.b, y.sub.b, z.sub.b].sup.T, w=[x, y, z].sup.T, and w.sub.kj=[x.sub.kj, y.sub.kj, 0].sup.T, and [x.sub.min, x.sub.max], [y.sub.min, y.sub.max], and [z.sub.min, z.sub.max] respectively represent candidate ranges along X, Y, and Z axes; n.sub.sCN (0,.sub.s.sup.2I.sub.M) represents amplified noise introduced at the MF-RIS with noise power .sub.s.sup.2 per unit, and n.sub.kjCN (0,.sub.u.sup.2) represents additive white Gaussian noise (AWGN) at the user U.sub.kj with power .sub.u.sup.2; a constant a.sub.m represents impact of a circuit sensitivity limitation on the m.sup.th element, and P.sub.b, P.sub.DC, P.sub.C respectively represent power consumed by each phase shifter, DC bias power consumed by the amplifier circuit, and power consumed by the RF-DC conversion circuit; and is a reciprocal of an energy conversion coefficient, and P.sub.O=.sub.k(.sub.kH.sub.kf.sub.k.sup.2+.sub.s.sup.2.sub.kI.sub.M.sup.2) represents output power of the MF-RIS.

4. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 3, wherein decomposing the non-convex problem into the three subproblems in S2 comprises: removing the constraint R.sub.l.fwdarw.j.sup.kR.sub.j.fwdarw.j.sup.k, kK, jJ.sub.k, IL.sub.k, from the non-convex problem; introducing a relaxation variable set .sub.0={A.sub.kj, B.sub.kj, .sub.kj, C.sub.m, .sub.m} such that: $A_{\mathrm{kj}}^{-1}={.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_k.Math.}^2,$ $B_{\mathrm{kj}}={.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_{\stackrel{}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2,$ ${}_{\mathrm{kj}}=A_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},$ ${.Math.}_m{C}_m^{-1}=\left(W+P_O\right)\left(1-\right)Z^{-1}+M,$ ${}_m=P_m^{\mathrm{RF}}$ rewriting the constraints as: $A_{\mathrm{kj}}^{-1}{.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_k.Math.}^2,$ $B_{\mathrm{kj}}{.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_{\stackrel{}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2,$ ${}_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},$ $kK,$ $j{J}_k,$ ${}_{\mathrm{kj}}{A}_{kl}^{-1}{B}_{kl}^{-1},$ $kK,$ $j{J}_k,$ $l{L}_k,$ $\left(W+P_O\right)\left(1-\right)Z^1+M{.Math.}_m{C}_m^{-1},$ ${}_m{P}_m^{\mathrm{RF}},$ $C_m\exp \left(-a\left({}_m-q\right)\right)+1,$ $m,$ wherein W=2(P.sub.b+P.sub.DC).sub.m.sub.m+(M.sub.m.sub.m)P.sub.C; and processing the non-convex constraints in the foregoing second formula through SCA; and performing first-order Taylor expansion to obtain lower bounds of right-hand terms of A.sub.kl.sup.1B.sub.kl.sup.1 and .sub.m C.sub.m.sup.1 at feasible points {, , } in a .sup.th iteration as follows: ${}_{kl}^{lb}=\frac{1}{A_{kl}^{\left(\right)}{B}_{kl}^{\left(\right)}}-\frac{A_{kl}-A_{kl}^{\left(\right)}}{{\left(A_{kl}^{\left(\right)}\right)}^2{B}_{kl}^{\left(\right)}}-\frac{B_{kl}-B_{kl}^{\left(\right)}}{{\left(B_{kl}^{\left(\right)}\right)}^2{A}_{kl}^{\left(\right)}},$ $C^{lb}={.Math.}_m\left(\frac{2}{C_m^{\left(\right)}}-\frac{C_m}{{\left(C_m^{\left(\right)}\right)}^2}\right)$ wherein in this way, the non-convex problem is decomposed into the three subproblems: the BS transmit beamforming optimization problem, the MF-RIS coefficient design problem, and the MF-RIS deployment optimization problem.

5. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 4, wherein in S3, for the BS transmit beamforming optimization problem, given {.sub.k, w}, auxiliary variables Q.sub.kj and C.sub.kj are introduced to transform an objective function $\underset{f_k,{}_{k}w}{\max }{.Math.}_k{.Math.}_{j{J}_k}{R}_{j.fwdarw.j}^k,Q_{\mathrm{kj}}=R_{j.fwdarw.j}^k$ and C.sub.kj=|h.sub.kjf.sub.k|.sup.2 P.sub.kj+B.sub.kj, the non-convex constraint Q.sub.kjlog.sub.2(1+p.sub.kjA.sub.kj.sup.1C.sub.kj.sup.1) is processed through SCA, a lower bound of a right-hand term of the non-convex constraint is obtained in the .sup.th iteration, H.sub.kj= and F.sub.k=k.sub.kf.sub.k.sup.H are defined, F.sub.k0 and rank(F.sub.k)=1, an auxiliary variable set ${}_1=\left\{A_{\mathrm{kj}},B_{\mathrm{kj}},C_{\mathrm{kj}},Q_{\mathrm{kj}},{}_{\mathrm{kj}},C_{\mathrm{kj}},{}_m\right\}$ $\mathrm{and}$ $\stackrel{}{W}=\frac{\left(C^{lb}-M\right)Z}{\left(1-\right)}-\frac{W}{}$ are introduced such that W.sub.k Tr(.sub.k(H(.sub.k F.sub.k)H.sup.H+.sub.s.sup.2I.sub.M).sub.k.sup.H), a rank-one constraint rank(F.sub.k)=1, k is replaced by a linear constraint ().sup.HTr(), k, through the SROCR method, [0, 1] is a trace ratio parameter of F.sub.k in a (1).sup.th iteration, is an eigenvector corresponding to a maximum eigenvalue of , is a solution of in the (1).sup.th iteration, and the BS transmit beamforming optimization problem is transformed into a convex semidefinite programming (SDP) problem.

6. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 5, wherein an algorithm for solving the convex SDP problem into which the BS transmit beamforming optimization problem is transformed comprises: initializing feasible points {F.sub.k.sup.(0), w.sub.k.sup.(0)} and a step .sub.1.sup.(0), setting an iteration index .sub.1=0, and repeating the following steps until a stop criterion is satisfied: if the convex SDP problem is solvable, solving the convex SDP problem to update , and updating =; otherwise, updating ${}_1^{\left({}_1+1\right)}=\frac{{}_1^{\left({}_1\right)}}{2};$ and updating $w_k^{\left({}_1+1\right)}=\min \left(1,\frac{{}_{\mathrm{ma}x}\left(F_k^{\left({}_1+1\right)}\right)}{\mathrm{Tr}\left(F_k^{\left({}_1+1\right)}\right)}+{}_1^{\left({}_1+1\right)}\right)$ and .sub.1=.sub.1+1, and ending the current iteration.

7. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 4, wherein in S3, for the MF-RIS coefficient design problem, given {f.sub.k, w}, H.sub.kj=[diag(g.sub.kj.sup.H)], v.sub.k=[.sub.1{square root over (.sub.1.sup.k)} e.sup.j.sup.1.sup.k, .sub.2{square root over (.sub.2.sup.k)} e.sup.j.sup.2.sup.k, . . . , .sub.M{square root over (.sub.M.sup.k)} e.sup.j.sup.M.sup.k].sup.H, and u.sub.k=[v.sub.k; 1] are defined; and U.sub.k=u.sub.ku.sub.k.sup.H is defined, U.sub.k0, rank(U.sub.k)=1, [U].sub.m,m=.sub.m.sup.2.sub.m.sup.k, [U.sub.k].sub.M+1, M+1=1, and the following equation is obtained: ${.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_{\mathrm{kj}}.Math.}^2={.Math.\left(h_{\mathrm{kj}}^H+g_{\mathrm{kj}}^H{}_{k}H\right)f_k.Math.}^2=\mathrm{Tr}\left(H_{\mathrm{kj}}{F}_k{H}_{\mathrm{kj}}^H{U}_k\right)$ similarly, |g.sub.kj.sup.H.sub.kn.sub.s|.sup.2=Tr(G.sub.kjU.sub.k) and P.sub.O=.sub.k Tr(HU.sub.k) are obtained; the constraints on A.sub.kj, B.sub.kj, C.sub.kj, W are rewritten as: $A_{\mathrm{kj}}^{-1}\mathrm{Tr}\left({\tilde{H}}_{\mathrm{kj}}{F}_k{\tilde{H}}_{\mathrm{kj}}^H{U}_k\right),$ $B_{\mathrm{kj}}\mathrm{Tr}\left(\left({\tilde{H}}_{\mathrm{kj}}{F}_{\stackrel{}{k}}{\tilde{H}}_{\mathrm{kj}}^H+{\tilde{G}}_{\mathrm{kj}}\right)U_k\right)+{}_u^2,$ $C_{\mathrm{kj}}\mathrm{Tr}\left({\tilde{H}}_{\mathrm{kj}}{F}_k{\tilde{H}}_{\mathrm{kj}}^H{U}_k\right)P_{\mathrm{kj}}+B_{\mathrm{kj},}$ $\stackrel{}{W}{.Math.}_k\mathrm{Tr}\left(\tilde{H}{U}_k\right)$ a rank-one constraint rank(U.sub.k)=1, k is approximated as ().sup.H Tr, k, and a binary constraint .sub.m{0, 1}, m is equivalently transformed into .sub.m.sub.m.sup.20, 0.sub.m1; and the constraint .sub.m.sub.m.sup.20 is processed through SCA; and an auxiliary variable .sub.m.sup.k=.sub.m.sup.2.sub.m.sup.k is introduced, and a non-convex constraint [U.sub.k].sub.m,m=.sub.m.sup.2.sub.m.sup.k, m,k is equivalently represented as [U.sub.k].sub.m,m=.sub.m.sup.k, .sub.m.sup.k=.sub.m.sup.2.sub.m.sup.k, and the equality constraint .sub.m.sup.k=.sub.m.sup.2.sub.m.sup.k is processed through a penalty function to transform the MF-RIS coefficient design problem into a penalty function method-based problem.

8. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 7, wherein an algorithm for solving the penalty function method-based problem comprises: initializing feasible points {U.sub.k.sup.(0), .sub.k.sup.(0)}>1, and a step .sub.2.sup.(0), and setting an iteration index .sub.22=0 and a maximum value of a penalty factor .sub.max; and repeating the following steps: if .sub.2.sub.2.sup.max and the penalty function method-based problem is solvable, solving the penalty function method-based problem to update , and updating =; otherwise, updating =; updating $v_k^{\left({}_2+1\right)}=\min \left(1,\frac{{}_{\mathrm{ma}x}\left(U_k^{\left({}_2+1\right)}\right)}{\mathrm{Tr}\left(U_k^{\left({}_2+1\right)}\right)}+{}_2^{\left({}_2+1\right)}\right);$ and if .sup.(.sup.2.sup.+1)=min{.sup.(.sup.2.sup.), .sub.max}, updating .sub.2=.sub.2=1 and ending the current iteration; otherwise, reinitializing U.sub.k.sup.(0) and letting >1 and .sub.2=0 until a stop criterion is satisfied.

9. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 4, wherein in S3, for the MF-RIS deployment optimization problem, the position w of the MF-RIS is designed through the local area optimization method, w.sup.(i1) is defined as a position of the MF-RIS obtained in an (i1).sup.th iteration, and a position variable satisfies a constraint ww.sup.(i1); it is assumed that .sup.(i1) and .sub.kj.sup.(i1) are obtained in the (i1).sup.th iteration, .sup.(i1) and .sub.kj.sup.(i1) are respectively an array response and small-scale fading after the (i1).sup.th iteration from the BS to the MF-RIS and from the MF-RIS to the user U.sub.kj, and a constraint comprising A.sub.kj, B.sub.kj, C.sub.kj, d.sub.bs.sup..sup.bs, d.sub.bs.sup..sup.bs is re-expressed; an auxiliary variable set is introduced to replace complex terms of the constraint, and a non-convex part of the constraint is approximated through SCA; and a right-hand term of a non-convex constraint r.sub.kjd.sub.kj.sup.T.sub.kjF.sub.k.sub.kj.sup.Hd.sub.kj is a convex term with respect to d.sub.kj.sup.T, the constraint r.sub.kjd.sub.kj.sup.T.sub.kjF.sub.k.sub.kj.sup.Hd.sub.kj is rewritten as a convex constraint r.sub.kj).sup.T.sub.kjF.sub.k.sub.kj.sup.H+2(().sup.T.sub.kj F.sub.k.sub.kj.sup.Hd.sub.kj), through first-order Taylor expansion, and is a feasible point in the .sup.th iteration.

10. The application method of the MF-RIS integrating signal reflection, refraction and amplification and energy harvesting in the multi-user wireless network according to claim 9, wherein an algorithm for solving the local area-based problem comprises: initializing feasible points {w.sup.(0), t.sup.(0), t.sub.kj.sup.(0), .sup.(0)} and setting an iteration index .sub.3=0; and repeating the following steps: solving the problem to update {, , , }, and updating .sub.3=.sub.3+1 until a stop criterion is satisfied.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) To describe the technical solutions in embodiments of the present application or in the prior art more clearly, the following briefly describes the accompanying drawings required for describing the embodiments. Apparently, the accompanying drawings in the following description show merely some embodiments of the present disclosure, and persons of ordinary skill in the art may still derive other accompanying drawings from these accompanying drawings.

(2) FIG. 1 is a schematic implementation diagram of an MF-RIS according to an embodiment of the present disclosure; and

(3) FIG. 2 is a flowchart of an AO algorithm according to an embodiment of the present disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(4) To better understand the technical solutions, the foregoing describes in detail a method in the present disclosure with reference to the accompanying drawings.

(5) Referring to FIG. 1, the present disclosure provides an MF-RIS integrating signal reflection, refraction and amplification and energy harvesting, having two working modes: an energy harvesting mode (H mode) and a signal relay mode (S mode). In the signal relay mode, an incident signal is reflected and refracted through surface equivalent electrical impedance and magnetoimpedance elements. The incident signal is divided into two parts by controlling electric current and magnetic current through an MCU chip. One part is reflected to reflection half-space and the other part is refracted to refraction half-space. A reflected signal and a refracted signal are amplified through an amplifier circuit. In the energy harvesting mode, RF energy is obtained from the incident signal and converted into DC power through an impedance matcher, an RF-DC conversion circuit and a capacitor, and an energy management module controls energy to be stored in an energy storage apparatus or supplied for operation of a phase shifter and the amplifier circuit. A circuit connection is adjusted such that each element is capable of being flexibly switched between the energy harvesting mode and the signal relay mode. Because a switching frequency of positive and negative intrinsic diodes is up to 5 MHz, a mode switching time of the MF-RIS is negligible compared with a typical wireless channel coherence time (for example, 4.2 ms).

(6) An energy harvesting circuit mainly relies on the following elements:

(7) An impedance matching network composed of resonators with a high quality factor ensures maximum power transmission from an element to a rectifier block. The RF-DC conversion circuit rectifies available RF power into DC voltage. The capacitor is configured to ensure electric current is smoothly transmitted to the energy storage apparatus or used as a short-term reserve when RF energy is unavailable. The power management module decides whether to store the energy obtained through conversion or use the energy for signal reflection, transmission, and amplification. The energy storage apparatus (such as a rechargeable battery and a supercapacitor) is configured to store the energy. When harvested energy exceeds consumed energy, excess energy is stored for future use, to achieve continuous self-sufficiency.

(8) For other elements working in the S mode, the incident signal is divided into the two parts by controlling the electric current and magnetic current. One part is reflected to the reflection half-space and the other part is refracted to the refraction half-space. With the help of the MCU, these elements can use the harvested energy to maintain the operation of the phase shifter and the amplifier circuit. Therefore, the provided MF-RIS does not require any external power supply in principle. Implementation of a reflect and transmit amplifier is also shown in FIG. 1. An operational amplifier-based electric current converter is configured to generate amplified reflect and transmit signals.

(9) An MF-RIS structure integrating energy harvesting and signal reflection, refraction and amplification provided in the present disclosure can implement signal reflection, transmission or refraction, and amplification and energy harvesting. An implementation of the MF-RIS is also provided, and an operation protocol in an actual wireless network is designed for the MF-RIS.

(10) Specifically, application of the MF-RIS provided in the present disclosure in a multi-user wireless network includes the following steps:

(11) S1: A mixed integer non-linear programming non-convex problem of jointly designing operation modes and parameters that include BS transmit beamforming, and different components and a deployment position of the MF-RIS is constructed to maximize an SR of a plurality of users.

(12) To characterize a signal model of the MF-RIS, it is considered that the MF-RIS has M elements. A set of the elements of the MF-RIS is indexed as M={1, . . . , M}. s.sub.m represents a signal received by the m.sup.th element. Due to hardware limitations, it is considered that each element cannot simultaneously work in both the H and S modes. Therefore, signals harvested, reflected, and refracted by the m.sup.th element are modeled as follows:

(13) $y_m^h=\left(1-{}_m\right)s_m,y_m^r={}_m\sqrt{{}_m}{e}^{j{}_m}{s}_m,y_m^t={}_m\sqrt{{}_m}{e}^{j{}_m}{s}_m$

(14) .sub.m={0, 1}, .sub.m.sup.r, .sub.m.sup.t[0, 2), .sub.m.sup.r, .sub.m.sup.t[0, .sub.max] respectively represent an energy harvesting coefficient, and reflection and refraction phase shifts and their corresponding amplitudes. .sub.m=1 represents that the m.sup.th element works in the S mode, and .sub.m=0 represents that the element works in the H mode. .sub.max1 represents an amplification factor. According to a law of conservation of energy, the amplifier should not consume more energy than maximum available energy that can be provided by the MF-RIS, that is, .sub.m.sup.r+B.sub.m.sup.t.sub.max. Reflection and refraction coefficients of the MF-RIS are modeled as follows:

(15) ${}_r=\mathrm{diag}\left({}_1\sqrt{{}_1^r}{e}^{j{}_1^r},{}_2\sqrt{{}_2^r}{e}^{j{}_2^r},.Math.,{}_M\sqrt{{}_M^r}{e}^{j{}_M^r}\right)$${}_t=\mathrm{diag}\left({}_1\sqrt{{}_1^t}{e}^{j{}_1^t},{}_2\sqrt{{}_2^t}{e}^{j{}_2^t},.Math.,{}_M\sqrt{{}_M^t}{e}^{j{}_M^t}\right)$${}_m\left\{0,1\right\},{}_m^r,{}_m^t\left[0,{}_{m\mathrm{ax}}\right],{}_m^r+{}_m^t{}_{m\mathrm{ax}},{}_m^r,{}_m^t[0,2)$

(16) An MF-RIS-assisted downlink NOMA network is considered. An N-antenna BS serves J single-antenna users with the help of an MF-RIS composed of M units. r(t) represents reflection (refraction) space. K={r,t} represents a space set, and J={1, 2, . . . , J} represents a user set. J.sub.k={1, 2, . . . , K.sub.k)} represents a user set in space K. J.sub.rJ.sub.t=J. For symbol simplicity, a user J in the space K is indexed as U.sub.kj.

(17) Considering a 3D Cartesian coordinate system, positions of the BS, the MF-RIS, and the user U.sub.kj are respectively w.sub.b=[x.sub.b, y.sub.b, z.sub.b].sup.T, w=[x, y, z].sup.T, and w.sub.kj=[x.sub.kj, y.sub.kj, 0].sup.T. Due to limited coverage of the MF-RIS, its deployable area is also limited. P represents a predefined deployment area of the MF-RIS, and the following constraint should be satisfied:

(18) $wP=\left\{{\left[x,y_{r}z\right]}^T|x_{min}x{x}_{\mathrm{ma}x},y_{min}y{y}_{max},z_{min}z{z}_{max}\right\},$

(19) [x.sub.min, x.sub.max], [y.sub.min, y.sub.max], and [z.sub.min, z.sub.max] respectively represent candidate ranges along X, Y, and Z axes.

(20) To characterize maximum performance that the MF-RIS can achieve, perfect channel state information for all channels is assumed to be available. Ricean fading modeling is performed for all channels. For example, a channel matrix H.sup.MN between the BS and the MF-RIS is as follows:

(21) $H=\underset{L_{\mathrm{bs}}}{\underset{}{\sqrt{h_0{d}_{\mathrm{bs}}^{-{}_{\mathrm{bs}}}}}}\underset{\hat{H}}{\underset{}{\left(\sqrt{\frac{{}_{\mathrm{bs}}}{{}_{\mathrm{bs}}+1}}{H}^{\mathrm{LoS}}+\sqrt{\frac{1}{{}_{\mathrm{bs}}+1}}{H}^{\mathrm{NLoS}}\right)}},$

(22) L.sub.bs is a distance-dependent path loss. constitutes an array response and small scale fading. Specifically, h.sub.0 is a path loss at a reference distance of 1 meter, d.sub.bs is a link distance between the BS and the MF-RIS, and .sub.bs is a corresponding path loss index. For small-scale fading, .sub.bs is a Ricean factor, and H.sup.NLoS is a non-line-of-sight (NLOS) component that follows independent and identically distributed Rayleigh fading. It is assumed that the MF-RIS is parallel to a Y-Z plane. The M elements of the MF-RIS form an uniform rectangular array M.sub.yM.sub.z=M. A line-of-sight (LOS) component H.sup.LoS is expressed as follows:

(23) $H^{\mathrm{LoS}}={\left[1,e^{-j\frac{2}{}\mathrm{dsin}{}_r\mathrm{si}n{}_r},.Math.,e^{-j\frac{2}{}\left(M_z-1\right)\mathrm{dsin}{}_{r}sin{}_r}\right]}^T$$.Math.{\left[1,e^{-j\frac{2}{}d\mathrm{co}s{}_{r}sin{}_r},.Math.,e^{-j\frac{2}{}\left(M_y-1\right)d\mathrm{co}s{}_{r}sin{}_r}\right]}^T$$.Math.\left[1,e^{-j\frac{2}{}d\mathrm{si}n{}_{r}cos{}_r},.Math.,e^{-j\frac{2}{}\left(N-1\right)d\mathrm{si}n{}_r\mathrm{co}s{}_r}\right],$

(24) An operator .Math. represents a Kronecker product, is a carrier wavelength, and d is an antenna distance. .sup.r, .sub.r, .sup.t, and .sub.t respectively represent vertical and horizontal angles of arrival, and vertical and horizontal angles of departure. A channel vector h.sub.kj.sup.H.sub.1N from the BS to the user U.sub.kj and a channel vector g.sub.kj.sup.H.sup.1M from the MF-RIS to the user U.sub.kj can be obtained through a procedure similar to that for obtaining H, and are as follows:

(25) $h_{\mathrm{kj}}=\underset{L_{\mathrm{bkj}}}{\underset{}{\sqrt{h_0{d}_{\mathrm{bkj}}^{-{}_{\mathrm{bkj}}}}}}\underset{{\hat{h}}_{\mathrm{kj}}}{\underset{}{\left(\sqrt{\frac{{}_{\mathrm{bkj}}}{{}_{\mathrm{bkj}}+1}}{h}_{\mathrm{bkj}}^{\mathrm{LoS}}+\sqrt{\frac{1}{{}_{\mathrm{bkj}}+1}}{h}_{\mathrm{bkj}}^{\mathrm{NLoS}}\right)}},$$g_{\mathrm{kj}}=\underset{L_{\mathrm{skj}}}{\underset{}{\sqrt{h_0{d}_{\mathrm{skj}}^{-{}_{\mathrm{skj}}}}}}\underset{{\hat{g}}_{\mathrm{kj}}}{\underset{}{\left(\sqrt{\frac{{}_{\mathrm{skj}}}{{}_{\mathrm{skj}}+1}}{g}_{\mathrm{kj}}^{\mathrm{LoS}}+\sqrt{\frac{1}{{}_{\mathrm{skj}}+1}}{g}_{\mathrm{kj}}^{\mathrm{NLoS}}\right)}}$

(26) To facilitate NOMA transmission, the BS transmits a superposed signal through a plurality of beamforming vectors, that is,

(27) $s=\underset{k}{.Math.}{f}_k\underset{j{J}_k}{.Math.}\sqrt{p_{\mathrm{kj}}{s}_{\mathrm{kj}}}.$
f.sub.k is a transmit beamforming vector of the space k. p.sub.kj is a power allocation factor for the user U.sub.kj. s.sub.kjCN (0,1) represents a corresponding modulated data symbol, which is independent of k. Therefore, a signal received at the user U.sub.kj is as follows:

(28) 0$y_{\mathrm{kj}}={\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k\sqrt{p_{\mathrm{kj}}{s}_{\mathrm{kj}}}+{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k{.Math.}_{i\left\{J_k/j\right\}}\sqrt{p_{\mathrm{ki}}{s}_{\mathrm{ki}}}+{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}{.Math.}_{i{J}_{\stackrel{\_}{k}}}\sqrt{p_{\stackrel{\_}{k}i}}{s}_{\stackrel{\_}{k}i}+g_{\mathrm{kj}}^H{}_k{n}_s+n_{\mathrm{kj}},$

(29) If k=t, k=r. If k=r, k=t. n.sub.sCN (0,.sub.s.sup.2I.sub.M) represents amplified noise introduced at the MF-RIS with noise power .sub.s.sup.2 per unit. n.sub.kjCN (0,.sub.u.sup.2) represents AWGN at the user U.sub.kj with noise power of .sub.u.sup.2. h.sub.kj=h.sub.kj.sup.H+g.sub.kj.sup.H.sub.kH represents a combined channel vector from the BS to the user U.sub.kj.

(30) According to a NOMA protocol, all users cancel interference through serial interference cancellation (SIC). It is assumed that equivalent combined channel gains of users in the space k in ascending order are expressed as follows:

(31) $\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2}\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2},kK,j{J}_k,l{L}_k,$

(32) L.sub.k={j, j+1, . . . , J.sub.k}. Therefore, for any users U.sub.kj and U.sub.kl satisfying jl, an achievable rate at which the user U.sub.kl decodes an expected signal of the user U.sub.kj is expressed as follows:

(33) $R_{l.fwdarw.j}^k={\log }_2\left(\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2{p}_{\mathrm{kj}}}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2{P}_{\mathrm{kj}}+{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2}\right),$

(34) $P_{\mathrm{kj}}=\underset{i=j+1}{\overset{J_k}{.Math.}}{p}_{\mathrm{ki}}.$

(35) To ensure that SIC is successful, an achievable signal to interference plus noise ratio (SINR) when the user U.sub.kl decodes the signal of the user U.sub.kj should not be less than an achievable SINR when the user U.sub.kj decodes its own signal, where jl. Therefore, there is the following SIC decoding rate constraint:

(36) $R_{l.fwdarw.j}^k{R}_{j.fwdarw.j}^k,kK,j{J}_k,l{L}_k$

(37) An energy harvesting coefficient matrix of the m.sup.th element is defined as

(38) $T_m=\mathrm{diag}\left(\left[\underset{1\mathrm{to}m-1}{\underset{}{0.Math.}}1-{}_m\underset{m+1\mathrm{to}M}{\underset{}{.Math.0}}\right]\right).$

(39) RF power received at the m.sup.th element is expressed as follows:

(40) $P_m^{\mathrm{RF}}=E\left({.Math.T_m\left(\mathrm{Hs}+n_s\right).Math.}^2\right),$

(41) To capture a dynamic change of RF energy conversion efficiency at different input power levels, a non-linear energy harvesting model is used in the present disclosure. Therefore, total power harvested by the m.sup.th element is expressed as follows:

(42) $P_m^A=\frac{Y_m-Z}{1-},Y_m=\frac{Z}{1+e^{-a\left(P_m^{\mathrm{RF}}-q\right)}},=\frac{1}{1+e^{\mathrm{aq}}},$

(43) Y.sub.m is a logical function of the received RF power P.sub.m.sup.RF. Z0 is a constant that determines maximum harvested power. A constant is used to ensure zero input/zero output response in the H mode. Constants a>0 and q>0 represent combined effects of a circuit sensitivity limitation and electric current leakage. To achieve self-sustainability of the MF-RIS, the following energy constraint should be satisfied:

(44) $2\left(P_b+P_{\mathrm{DC}}\right){.Math.}_m{}_m+\left(M-{.Math.}_m{}_m\right)P_C+P_O{.Math.}_m{P}_m^A,$

(45) P.sub.b, P.sub.DC, P.sub.C respectively represent power consumed by each phase shifter, DC bias power consumed by the amplifier circuit, and power consumed by the RF-DC conversion circuit. is a reciprocal of an energy conversion coefficient. P.sub.O=.sub.k(.sub.kH.sub.kf.sub.k.sup.2+.sub.s.sup.2.sub.kI.sub.M.sup.2) represents output power of the MF-RIS.

(46) An objective is to maximize the achievable SR of all users by jointly optimizing power allocation, BS transmit beamforming, the coefficient matrix, and a 3D position of the MF-RIS while maintaining self-sustainability of the MF-RIS. The following optimization problem is constructed:

(47) $\underset{f_k,{}_k,w}{\max }{.Math.}_k{.Math.}_{j{J}_k}{R}_{j.fwdarw.j}^k$$s.t.{.Math.}_k{.Math.f_k.Math.}^2{P}_{\mathrm{BS}}^{\max },$${}_k{R}_{\mathrm{MF}},kK,$$R_{j.fwdarw.j}^k{R}_{\mathrm{kj}}^{\min },kK,j{J}_k,$$wP=\left\{{\left[x,y,z\right]}^T|x_{\min }x{x}_{\max },y_{\min }y{y}_{\max },z_{\min }z{z}_{\max }\right\},$$\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2}\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2},$$kK,j{J}_k,l{L}_k,$$R_{l.fwdarw.j}^k{R}_{j.fwdarw.j}^k,kK,j{J}_k,l{L}_k$$2\left(P_b+P_{\mathrm{DC}}\right){.Math.}_m{}_m\left(M-{.Math.}_m{}_m\right)P_C+P_O{.Math.}_m{P}_m^A,$

(48) P.sub.BS.sup.max represents maximum transmit power of the BS. R.sub.kj.sup.min represents a minimum quality of service requirement of the user U.sub.kj.

(49) R.sub.MF={.sub.m, .sub.m.sup.k, .sub.m.sup.k|.sub.m{0, 1}, .sub.m.sup.k[0, .sub.max], .sub.k.sub.m.sup.k.sub.max, .sub.m.sup.k[0, 2), m, k} is a feasible coefficient set of the MF-RIS. The constraint .sub.kf.sub.k.sup.2O.sub.BS.sup.max limits total transmit power of the BS.

(50) S2: The non-convex problem constructed in S1 is transformed into a more tractable form, and an AO-based algorithm is proposed to effectively find a high-performance suboptimal solution. The non-convex problem is decomposed into three subproblems: a BS transmit beamforming optimization problem, an MF-RIS coefficient design problem, and an MF-RIS deployment optimization problem.

(51) Before the original problem is solved, the original problem is transformed into the more tractable form. First, the constraint R.sub.l.fwdarw.j.sup.kR.sub.j.fwdarw.j.sup.k, kK, jJ.sub.k, lL.sub.k is a necessary condition of the following inequality:

(52) 0$\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2}\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2},$
kK, jJ.sub.k, lL.sub.k,

(53) Specifically, according to the foregoing inequality, the equivalent combined channel gains of the users U.sub.kj and U.sub.kl whose decoding orders jl satisfy the following condition:

(54) ${.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2\left({.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2\right){.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2\left({.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2\right)$

(55) Both sides of the inequality are multiplied by p.sub.kj, and p.sub.kj|h.sub.klf.sub.2|.sup.2|h.sub.kjf.sub.k|.sup.2 P.sub.kj is added to both sides. An equivalent transformation is performed to obtain the following inequality:

(56) $\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k.Math.}^2{p}_{\mathrm{kj}}}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2{P}_{\mathrm{kj}}+{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2}\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2{p}_{\mathrm{kj}}}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2{P}_{\mathrm{kj}}+{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2}$

(57) Apparently, the foregoing inequality ensures that the constraint R.sub.l.fwdarw.j.sup.kR.sub.j.fwdarw.j.sup.k, kK, jJ.sub.k, lL.sub.k is satisfied. Therefore, when the constraint

(58) $\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2}\frac{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2}{{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2},$
kK, jJ.sub.k, lL.sub.k exists, removing the constraint R.sub.l.fwdarw.j.sup.kR.sub.j.fwdarw.j.sup.k, kK, jJ.sub.k, lL.sub.k does not affect optimality of the original problem. Therefore, the constraint R.sub.l.fwdarw.j.sup.kR.sub.j.fwdarw.j.sup.k, kK, jJ.sub.k, lL.sub.k can be removed from the original problem.

(59) Next, to process the following highly coupled constraints:

(60) ${.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_k.Math.}^2\left({.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2\right){.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2\left({.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2\right)$$2\left(P_b+P_{\mathrm{DC}}\right){.Math.}_m{}_m\left(M-{.Math.}_m{}_m\right)P_C+P_O{.Math.}_m{P}_m^A$

(61) A relaxation variable set .sub.0={A.sub.kj, B.sub.kj, .sub.kj, C.sub.m, .sub.m} is introduced such that:

(62) $A_{\mathrm{kj}}^{-1}={.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2,B_{\mathrm{kj}}={.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2,$${}_{\mathrm{kj}}=A_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},$${.Math.}_m{C}_m^{-1}\left(W+P_O\right)\left(1-\right)Z^{-1}+M,$${}_m=P_m^{\mathrm{RF}}$

(63) With these variable definitions, the foregoing two constraints are rewritten as follows:

(64) $A_{\mathrm{kj}}^{-1}{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2,B_{\mathrm{kj}}{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2,$${}_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},kK,j{J}_k,$${}_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},kK,j{J}_k,l{L}_k,$$\left(W+P_O\right)\left(1-\right)Z^{-1}+M{.Math.}_m{C}_m^{-1},$${}_m{P}_m^{\mathrm{RF}},C_m\exp \left(-a\left({}_m-q\right)\right)+1,m,$

(65) W=2(P.sub.b+P.sub.DC).sub.m.sub.m+(M.sub.m.sub.m)P.sub.C. The constraints in the foregoing second formula are non-convex because right-hand terms of the constraints are convex. The constraints are processed through SCA. Based on a fact that first-order Taylor expansion of a convex function is a global underestimation measure, lower bounds of the right-hand terms at feasible points {, , } in an .sup.th iteration are expressed as follows:

(66) ${}_{\mathrm{kl}}^{1b}=\frac{1}{A_{\mathrm{kl}}^{\left(\right)}{B}_{\mathrm{kl}}^{\left(\right)}}-\frac{A_{\mathrm{kl}}-A_{\mathrm{kl}}^{\left(\right)}}{{\left(A_{\mathrm{kl}}^{\left(\right)}\right)}^2{B}_{\mathrm{kl}}^{\left(\right)}}-\frac{B_{\mathrm{kl}}-B_{\mathrm{kl}}^{\left(\right)}}{{\left(B_{\mathrm{kl}}^{\left(\right)}\right)}^2{A}_{\mathrm{kl}}^{\left(\right)}},$$C^{1b}={.Math.}_m\left(\frac{2}{C_m^{\left(\right)}}-\frac{C_m}{{\left(C_m^{\left(\right)}\right)}^2}\right)$

(67) As a result, the original problem is equivalently transformed into the following problems:

(68) $\underset{f_k,{}_k,w,{}_0}{\max }{.Math.}_k{.Math.}_{j{J}_k}{R}_{j.fwdarw.j}^k$$s.t.{}_{\mathrm{kl}}{}_{\mathrm{kl}}^{1b},kK,j{J}_k,l{L}_k,$$\left(W+P_O\right)\left(1-\right)Z^{-1}+M{C}^{1b},$$wP=\left\{{\left[x,y,z\right]}^T|x_{\min }x{x}_{\max },y_{\min }y{y}_{\max },z_{\min }z{z}_{\max }\right\},$${.Math.}_k{.Math.f_k.Math.}^2{P}_{\mathrm{BS}}^{\max },$${}_k{R}_{\mathrm{MF}},kK,$$R_{j.fwdarw.j}^k{R}_{\mathrm{kj}}^{\min },kK,j{J}_k,$$A_{\mathrm{kj}}^{-1}{.Math.{\stackrel{\_}{h}}_{\mathrm{kl}}{f}_k.Math.}^2,B_{\mathrm{kj}}{.Math.{\stackrel{\_}{h}}_{\mathrm{kj}}{f}_{\stackrel{\_}{k}}.Math.}^2+{.Math.g_{\mathrm{kl}}^H{}_k{n}_s.Math.}^2+{}_u^2,$${}_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},kK,j{J}_k,$${}_m{P}_m^{\mathrm{RF}},C_m\exp \left(-a\left({}_m-q\right)\right)+1,m,$

(69) Then, the three subproblems are solved one by one.

(70) S3: For the BS transmit beamforming optimization problem in S2, auxiliary variables are introduced and the BS transmit beamforming optimization problem is solved through an SROCR method.

(71) First, for the BS transmit beamforming optimization problem, given {.sub.k, w}, an objective is to solve the transmit beamforming vector f.sub.k. Due to the non-concave objective function

(72) $\underset{f_k,{}_k,w}{\max }{.Math.}_k{.Math.}_{j{J}_k}{R}_{j.fwdarw.j}^k$
and the non-convex constraint R.sub.j.fwdarw.j.sup.kR.sub.kj.sup.min, kK, jJ.sub.k, the original problem is still difficult to be directly solved. In view of this, auxiliary variables Q.sub.kj and C.sub.kj are introduced, where Q.sub.kj=R.sub.j.fwdarw.j.sup.k and C.sub.kj|h.sub.kjf.sub.k|.sup.2 P.sub.kj+B.sub.kj. The objective function

(73) 0$\underset{f_k,{}_k,w}{\max }{.Math.}_k{.Math.}_{j{J}_k}{R}_{j.fwdarw.j}^k$
is transformed into:

(74) ${.Math.}_k{.Math.}_{j{J}_k}{R}_{j.fwdarw.j}^k={.Math.}_k{.Math.}_{j{J}_k}{Q}_{\mathrm{kj}}$

(75) In addition, the following new constraints are obtained:

(76) $C_{\mathrm{kj}}{.Math.h_{\mathrm{kj}}{f}_k.Math.}^2{P}_{\mathrm{kj}}+B_{\mathrm{kj}},$$Q_{\mathrm{kj}}{\log }_2\left(1+p_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{C}_{\mathrm{kj}}^{-1}\right),$$Q_{\mathrm{kj}}{R}_{\mathrm{kj}}^{\min }$

(77) The following non-convex constraint is processed through SCA:

(78) $Q_{\mathrm{kj}}{\log }_2\left(1+p_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{C}_{\mathrm{kj}}^{-1}\right)$

(79) Specifically, in the .sup.th iteration, a lower bound of a right-hand term of the constraint is expressed as follows:

(80) $R_{\mathrm{kj}}^{\mathrm{Ib}}={\log }_2\left(1+\frac{p_{\mathrm{kj}}}{A_{\mathrm{kj}}^{\left(\right)}{C}_{\mathrm{kj}}^{\left(\right)}}\right)-\frac{p_{\mathrm{kj}}\left({\log }_{2}e\right)\left(A_{\mathrm{kj}}-A_{\mathrm{kj}}^{\left(\right)}\right)}{p_{\mathrm{kj}}{A}_{\mathrm{kj}}^{\left(\right)}+{\left(A_{\mathrm{kj}}^{\left(\right)}\right)}^2{C}_{\mathrm{kj}}^{\left(\right)}}-\frac{p_{\mathrm{kj}}\left({\log }_{2}e\right)\left(C_{\mathrm{kj}}-C_{\mathrm{kj}}^{\left(\right)}\right)}{p_{\mathrm{kj}}{C}_{\mathrm{kj}}^{\left(\right)}+{\left(C_{\mathrm{kj}}^{\left(\right)}\right)}^2{A}_{\mathrm{kj}}^{\left(\right)}}$

(81) Next, H.sub.kj= and F.sub.k=f.sub.kf.sub.k.sup.H are defined. F.sub.k0 and rank(F.sub.k)=1. Then, the transmit beamforming vector is optimized by solving the following problem:

(82) $\underset{F_k,{}_1}{\max }{.Math.}_k{.Math.}_{j{J}_k}{Q}_{\mathrm{kj}}$$s.t.\mathrm{rank}\left(F_k\right)=1,k,$${.Math.}_k\mathrm{Tr}\left(F_k\right)P_{BS}^{\max },F_{k}0,k,$$A_{\mathrm{kj}}^{-1}\mathrm{Tr}\left({\stackrel{}{H}}_{\mathrm{kj}},F_k\right),B_{\mathrm{kj}},\mathrm{Tr}\left({\stackrel{}{H}}_{\mathrm{kj}}{F}_{\stackrel{}{k}}\right)+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2,kK,j{J}_k,$$C_{\mathrm{kj}}\mathrm{Tr}\left({\stackrel{}{H}}_{\mathrm{kj}}{F}_k\right)P_{\mathrm{kj}}+B_{\mathrm{kj}},kK,j{J}_k,$${}_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},{}_{\mathrm{kj}}{}_{\mathrm{kl}}^{\mathrm{lb}},Q_{\mathrm{kj}}{R}_{\mathrm{kj}}^{\mathrm{lb}},Q_{\mathrm{kj}}{R}_{\mathrm{kj}}^{\min },kK,j{J}_k,l{L}_k,$$C_m\exp \left(-a\left({}_m-q\right)\right)+1,m,$$\stackrel{}{W}{.Math.}_k\mathrm{Tr}\left({}_k\left(H\left({.Math.}_k{F}_k\right)H^H+{}_s^2{I}_M\right){}_k^H\right),$${}_m\mathrm{Tr}\left(T_{m}H\left({.Math.}_k{F}_k\right)H^H{T}_m^H\right)+{}_s^2\left(1-{}_m\right)m,$

(83) An auxiliary variable set .sub.1={A.sub.kj, B.sub.kj, C.sub.kj, Q.sub.kj, .sub.kj, C.sub.m, .sub.m} and

(84) $\stackrel{}{W}=\frac{\left({\mathrm{Cb}}^{lb}-M\right)Z}{\left(1-\right)}-\frac{W}{}$
are used. The main difficulty in solving the foregoing problem lies in the rank-one constraint rank(F.sub.k)=1, k. The constraint is processed through the SROCR method. A basic idea of the SROCR method is to gradually relax the rank-one constraint to find a feasible rank-one solution.

(85) Specifically, w.sub.k.sup.(l1)[0, 1] is defined as a trace ratio parameter of F.sub.k in a (1).sup.th iteration. The rank-one constraint rank(F.sub.k)=1, k in the .sup.th iteration may be replaced by the following linear constraint:

(86) ${\left(f_k^{e,\left(-1\right)}\right)}^H{F}_k^{\left(\right)}{f}_k^{e,\left(-1\right)}{w}_k^{\left(-1\right)}\mathrm{Tr}\left(F_k^{\left(\right)}\right),k,$

(87) is an eigenvector corresponding to a maximum eigenvalue of . is a solution of given in the (1).sup.th iteration. Therefore, the problem is transformed into:

(88) $\underset{F_k,{}_1}{\max }{.Math.}_k{.Math.}_{j{J}_k}{Q}_{\mathrm{kj}}$$s.t{.Math.}_k\mathrm{Tr}\left(F_k\right)P_{BS}^{\max },F_{k}0,k,$$A_{\mathrm{kj}}^{-1}\mathrm{Tr}\left({\stackrel{}{H}}_{\mathrm{kj}}{F}_k\right),B_{\mathrm{kj}}\mathrm{Tr}\left({\stackrel{}{H}}_{\mathrm{kj}}{F}_{\stackrel{}{k}}\right)+{.Math.g_{\mathrm{kj}}^H{}_k{n}_s.Math.}^2+{}_u^2,kK,j{J}_k,$$C_{\mathrm{kj}}\mathrm{Tr}\left({\stackrel{}{H}}_{\mathrm{kj}}{F}_k\right)P_{\mathrm{kj}}+B_{\mathrm{kj}},kK,j{J}_k,$${}_{\mathrm{kj}}{A}_{\mathrm{kj}}^{-1}{B}_{\mathrm{kj}}^{-1},{}_{\mathrm{kj}}{}_{\mathrm{kl}}^{\mathrm{lb}},Q_{\mathrm{kj}}{R}_{\mathrm{kj}}^{\mathrm{lb}},Q_{\mathrm{kj}}{R}_{\mathrm{kj}}^{\min },kK,j{J}_k,l{L}_k,$$C_m\exp \left(-a\left({}_m-q\right)\right)+1,m,$$\stackrel{}{W}{.Math.}_k\mathrm{Tr}\left({}_k\left(H\left({.Math.}_k{F}_k\right)H^H+{}_s^2{I}_M\right){}_k^H\right),$${}_m\mathrm{Tr}\left(T_{m}H\left({.Math.}_k{F}_k\right)H^H{T}_m^H\right)+{}_s^2\left(1-{}_m\right)m,$${\left(f_k^{e,\left(-1\right)}\right)}^H{F}_k^{\left(\right)}{f}_k^{e,\left(-1\right)}{w}_k^{\left(-1\right)}\mathrm{Tr}\left(F_k^{\left(\right)}\right),k,$

(89) The problem is an SDP problem and can be effectively solved through CVX. The rank-one solution is gradually approached by iteratively increasing from 0 to 1. The following describes an iterative algorithm for solving the problem. After the problem is solved, Cholesky decomposition is performed on F.sub.k to obtain a solution of f.sub.k, that is, =f.sub.kf.sub.k.sup.H.

(90) The algorithm for solving the problem through the SROCR method includes: Initialize feasible points {F.sub.k.sup.(0), w.sub.k.sup.(0)} and a step .sub.1.sup.(0). Set an iteration index .sub.1=0. Repeat the following steps until a stop criterion is satisfied: If the foregoing SDP problem is solvable, solve the problem to update , and update =; otherwise, update

(91) ${}_1^{\left({}_1+1\right)}=\frac{{}_1^{\left({}_1\right)}}{2}.$
Update

(92) 0$w_k^{\left({}_1+1\right)}=\min \left(1,\frac{{}_{\max }\left(F_k^{\left({}_1+1\right)}\right)}{\mathrm{Tr}\left(F_k^{\left({}_1+1\right)}\right)}+{}_1^{\left({}_1+1\right)}\right)$
and .sub.1=.sub.1+1, and end the current iteration.

(93) S4: For the MF-RIS coefficient design problem in S2, an auxiliary variable is introduced, a non-convex objective function is replaced by its CUB, an equality constraint is processed through a penalty function method, and the coefficient of the MF-RIS is designed.

(94) For any given {f.sub.k, w}, v.sub.k=[.sub.1{square root over (.sub.1.sup.k)} e.sup.j.sup.1.sup.k, .sub.2{square root over (.sub.2.sup.k)} e.sup.j.sup.2.sup.k, . . . , .sub.M{square root over (.sub.M.sup.k)} e.sup.j.sup.M.sup.k].sup.H is used to represent {tilde over (H)}.sub.kj=[diag(g.sub.kj.sup.H)H; h.sub.kj.sup.H], and u.sub.k=[vx.sub.k; 1]. U.sub.k=u.sub.ku.sub.k.sup.H is further defined. U.sub.k0, rank(U.sub.k)=1 and [U.sub.k].sub.m,m=.sub.m.sup.2.sub.m.sup.k, [U.sub.k].sub.M+1, M+1=1. The following equation holds:

(95) ${.Math.{\stackrel{}{h}}_{\mathrm{kj}}{f}_{\mathrm{kj}}.Math.}^2={.Math.\left(h_{\mathrm{kj}}^H+g_{\mathrm{kj}}^H{}_{k}H\right)f_k.Math.}^2=\mathrm{Tr}\left({\tilde{H}}_{\mathrm{kj}}{F}_k{\tilde{H}}_{\mathrm{kj}}^H{U}_k\right)$

(96) Similarly, the following equations are obtained:

(97) $|g_{\mathrm{kj}}^H{}_k{n}_s{|}^2=\mathrm{Tr}\left({\tilde{G}}_{\mathrm{kj}}{U}_k\right)\mathrm{and}$$P_O={.Math.}_k\mathrm{Tr}\left(\tilde{H}{U}_k\right)$$\mathrm{where}$${\tilde{G}}_{\mathrm{kj}}={\tilde{g}}_{\mathrm{kj}}{\tilde{g}}_{\mathrm{kj}}^H,\tilde{H}=\tilde{h}{\tilde{h}}^H+{}_s^2{\tilde{I}}_M{\tilde{I}}_M^H$${\tilde{g}}_{\mathrm{kj}}=\left[\mathrm{diag}\left(g_{\mathrm{kj}}^H\right)n_s;0\right],\tilde{h}=\left[\mathrm{Hs};0\right],{\tilde{I}}_M=\left[I_M;0_{1M}\right]$